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G = C23.615C24  order 128 = 27

332nd central stem extension by C23 of C24

p-group, metabelian, nilpotent (class 2), monomial

Aliases: C23.615C24, C24.414C23, C22.2902- 1+4, C22.3892+ 1+4, C22⋊C4.18D4, C23.72(C2×D4), C2.67(D4⋊6D4), C2.120(D4⋊5D4), C23.Q8⋊69C2, C23.7Q8⋊97C2, C23.11D4⋊96C2, (C22×C4).193C23, (C2×C42).666C22, (C23×C4).470C22, C23.8Q8⋊116C2, C22.424(C22×D4), C23.23D4.55C2, C23.10D4.46C2, (C22×D4).249C22, (C22×Q8).192C22, C24.C22⋊142C2, C23.67C23⋊87C2, C23.78C23⋊48C2, C2.73(C22.32C24), C23.65C23⋊128C2, C23.63C23⋊142C2, C2.C42.321C22, C2.80(C22.36C24), C2.56(C22.50C24), C2.18(C22.57C24), C2.51(C22.31C24), (C2×C4).426(C2×D4), (C2×C42⋊2C2)⋊23C2, (C2×C4).433(C4○D4), (C2×C4⋊C4).428C22, (C2×C4.4D4).32C2, C22.477(C2×C4○D4), (C2×C22⋊C4).280C22, SmallGroup(128,1447)

Series: Derived ►Chief ►Lower central ►Upper central ►Jennings

Derived series
C1 — C23 — C23.615C24
Chief series
C1 — C2 — C22 — C23 — C22×C4 — C2×C42 — C23.67C23 — C23.615C24
Lower central
C1 — C23 — C23.615C24
Upper central
C1 — C23 — C23.615C24
Jennings
C1 — C23 — C23.615C24

Generators and relations for C23.615C24
 G = < a,b,c,d,e,f,g | a2=b2=c2=d2=1, e2=g2=b, f2=cb=bc, eae-1=ab=ba, faf-1=ac=ca, ad=da, gag-1=abc, bd=db, geg-1=be=eb, bf=fb, bg=gb, cd=dc, ce=ec, cf=fc, cg=gc, fef-1=de=ed, df=fd, dg=gd, fg=gf >

Subgroups: 500 in 249 conjugacy classes, 96 normal (82 characteristic)
C1, C2, C2, C4, C22, C22, C2×C4, C2×C4, D4, Q8, C23, C23, C23, C42, C22⋊C4, C22⋊C4, C4⋊C4, C22×C4, C22×C4, C2×D4, C2×Q8, C24, C2.C42, C2×C42, C2×C22⋊C4, C2×C4⋊C4, C4.4D4, C42⋊2C2, C23×C4, C22×D4, C22×Q8, C23.7Q8, C23.8Q8, C23.23D4, C23.63C23, C24.C22, C23.65C23, C23.67C23, C23.10D4, C23.78C23, C23.Q8, C23.11D4, C2×C4.4D4, C2×C42⋊2C2, C23.615C24
Quotients: C1, C2, C22, D4, C23, C2×D4, C4○D4, C24, C22×D4, C2×C4○D4, 2+ 1+4, 2- 1+4, C22.31C24, C22.32C24, C22.36C24, D4⋊5D4, D4⋊6D4, C22.50C24, C22.57C24, C23.615C24

Smallest permutation representation of C23.615C24
►On 64 points
Generators in S64
(1 4)(2 3)(5 49)(6 52)(7 51)(8 50)(9 12)(10 11)(13 44)(14 43)(15 42)(16 41)(17 64)(18 63)(19 62)(20 61)(21 39)(22 38)(23 37)(24 40)(25 28)(26 27)(29 58)(30 57)(31 60)(32 59)(33 47)(34 46)(35 45)(36 48)(53 56)(54 55)

(1 3)(2 4)(5 7)(6 8)(9 11)(10 12)(13 15)(14 16)(17 19)(18 20)(21 23)(22 24)(25 27)(26 28)(29 31)(30 32)(33 35)(34 36)(37 39)(38 40)(41 43)(42 44)(45 47)(46 48)(49 51)(50 52)(53 55)(54 56)(57 59)(58 60)(61 63)(62 64)

(1 9)(2 10)(3 11)(4 12)(5 39)(6 40)(7 37)(8 38)(13 41)(14 42)(15 43)(16 44)(17 45)(18 46)(19 47)(20 48)(21 49)(22 50)(23 51)(24 52)(25 55)(26 56)(27 53)(28 54)(29 59)(30 60)(31 57)(32 58)(33 62)(34 63)(35 64)(36 61)

(1 53)(2 54)(3 55)(4 56)(5 52)(6 49)(7 50)(8 51)(9 27)(10 28)(11 25)(12 26)(13 31)(14 32)(15 29)(16 30)(17 36)(18 33)(19 34)(20 35)(21 40)(22 37)(23 38)(24 39)(41 57)(42 58)(43 59)(44 60)(45 61)(46 62)(47 63)(48 64)

(1 2 3 4)(5 6 7 8)(9 10 11 12)(13 14 15 16)(17 18 19 20)(21 22 23 24)(25 26 27 28)(29 30 31 32)(33 34 35 36)(37 38 39 40)(41 42 43 44)(45 46 47 48)(49 50 51 52)(53 54 55 56)(57 58 59 60)(61 62 63 64)

(1 19 11 45)(2 35 12 62)(3 17 9 47)(4 33 10 64)(5 44 37 14)(6 57 38 29)(7 42 39 16)(8 59 40 31)(13 51 43 21)(15 49 41 23)(18 28 48 56)(20 26 46 54)(22 32 52 60)(24 30 50 58)(25 61 53 34)(27 63 55 36)

(1 43 3 41)(2 42 4 44)(5 62 7 64)(6 61 8 63)(9 15 11 13)(10 14 12 16)(17 23 19 21)(18 22 20 24)(25 31 27 29)(26 30 28 32)(33 37 35 39)(34 40 36 38)(45 51 47 49)(46 50 48 52)(53 59 55 57)(54 58 56 60)
 
G:=sub<Sym(64)| (1,4)(2,3)(5,49)(6,52)(7,51)(8,50)(9,12)(10,11)(13,44)(14,43)(15,42)(16,41)(17,64)(18,63)(19,62)(20,61)(21,39)(22,38)(23,37)(24,40)(25,28)(26,27)(29,58)(30,57)(31,60)(32,59)(33,47)(34,46)(35,45)(36,48)(53,56)(54,55), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,19,11,45)(2,35,12,62)(3,17,9,47)(4,33,10,64)(5,44,37,14)(6,57,38,29)(7,42,39,16)(8,59,40,31)(13,51,43,21)(15,49,41,23)(18,28,48,56)(20,26,46,54)(22,32,52,60)(24,30,50,58)(25,61,53,34)(27,63,55,36), (1,43,3,41)(2,42,4,44)(5,62,7,64)(6,61,8,63)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32)(33,37,35,39)(34,40,36,38)(45,51,47,49)(46,50,48,52)(53,59,55,57)(54,58,56,60)>;
 
G:=Group( (1,4)(2,3)(5,49)(6,52)(7,51)(8,50)(9,12)(10,11)(13,44)(14,43)(15,42)(16,41)(17,64)(18,63)(19,62)(20,61)(21,39)(22,38)(23,37)(24,40)(25,28)(26,27)(29,58)(30,57)(31,60)(32,59)(33,47)(34,46)(35,45)(36,48)(53,56)(54,55), (1,3)(2,4)(5,7)(6,8)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,23)(22,24)(25,27)(26,28)(29,31)(30,32)(33,35)(34,36)(37,39)(38,40)(41,43)(42,44)(45,47)(46,48)(49,51)(50,52)(53,55)(54,56)(57,59)(58,60)(61,63)(62,64), (1,9)(2,10)(3,11)(4,12)(5,39)(6,40)(7,37)(8,38)(13,41)(14,42)(15,43)(16,44)(17,45)(18,46)(19,47)(20,48)(21,49)(22,50)(23,51)(24,52)(25,55)(26,56)(27,53)(28,54)(29,59)(30,60)(31,57)(32,58)(33,62)(34,63)(35,64)(36,61), (1,53)(2,54)(3,55)(4,56)(5,52)(6,49)(7,50)(8,51)(9,27)(10,28)(11,25)(12,26)(13,31)(14,32)(15,29)(16,30)(17,36)(18,33)(19,34)(20,35)(21,40)(22,37)(23,38)(24,39)(41,57)(42,58)(43,59)(44,60)(45,61)(46,62)(47,63)(48,64), (1,2,3,4)(5,6,7,8)(9,10,11,12)(13,14,15,16)(17,18,19,20)(21,22,23,24)(25,26,27,28)(29,30,31,32)(33,34,35,36)(37,38,39,40)(41,42,43,44)(45,46,47,48)(49,50,51,52)(53,54,55,56)(57,58,59,60)(61,62,63,64), (1,19,11,45)(2,35,12,62)(3,17,9,47)(4,33,10,64)(5,44,37,14)(6,57,38,29)(7,42,39,16)(8,59,40,31)(13,51,43,21)(15,49,41,23)(18,28,48,56)(20,26,46,54)(22,32,52,60)(24,30,50,58)(25,61,53,34)(27,63,55,36), (1,43,3,41)(2,42,4,44)(5,62,7,64)(6,61,8,63)(9,15,11,13)(10,14,12,16)(17,23,19,21)(18,22,20,24)(25,31,27,29)(26,30,28,32)(33,37,35,39)(34,40,36,38)(45,51,47,49)(46,50,48,52)(53,59,55,57)(54,58,56,60) );
 
G=PermutationGroup([[(1,4),(2,3),(5,49),(6,52),(7,51),(8,50),(9,12),(10,11),(13,44),(14,43),(15,42),(16,41),(17,64),(18,63),(19,62),(20,61),(21,39),(22,38),(23,37),(24,40),(25,28),(26,27),(29,58),(30,57),(31,60),(32,59),(33,47),(34,46),(35,45),(36,48),(53,56),(54,55)], [(1,3),(2,4),(5,7),(6,8),(9,11),(10,12),(13,15),(14,16),(17,19),(18,20),(21,23),(22,24),(25,27),(26,28),(29,31),(30,32),(33,35),(34,36),(37,39),(38,40),(41,43),(42,44),(45,47),(46,48),(49,51),(50,52),(53,55),(54,56),(57,59),(58,60),(61,63),(62,64)], [(1,9),(2,10),(3,11),(4,12),(5,39),(6,40),(7,37),(8,38),(13,41),(14,42),(15,43),(16,44),(17,45),(18,46),(19,47),(20,48),(21,49),(22,50),(23,51),(24,52),(25,55),(26,56),(27,53),(28,54),(29,59),(30,60),(31,57),(32,58),(33,62),(34,63),(35,64),(36,61)], [(1,53),(2,54),(3,55),(4,56),(5,52),(6,49),(7,50),(8,51),(9,27),(10,28),(11,25),(12,26),(13,31),(14,32),(15,29),(16,30),(17,36),(18,33),(19,34),(20,35),(21,40),(22,37),(23,38),(24,39),(41,57),(42,58),(43,59),(44,60),(45,61),(46,62),(47,63),(48,64)], [(1,2,3,4),(5,6,7,8),(9,10,11,12),(13,14,15,16),(17,18,19,20),(21,22,23,24),(25,26,27,28),(29,30,31,32),(33,34,35,36),(37,38,39,40),(41,42,43,44),(45,46,47,48),(49,50,51,52),(53,54,55,56),(57,58,59,60),(61,62,63,64)], [(1,19,11,45),(2,35,12,62),(3,17,9,47),(4,33,10,64),(5,44,37,14),(6,57,38,29),(7,42,39,16),(8,59,40,31),(13,51,43,21),(15,49,41,23),(18,28,48,56),(20,26,46,54),(22,32,52,60),(24,30,50,58),(25,61,53,34),(27,63,55,36)], [(1,43,3,41),(2,42,4,44),(5,62,7,64),(6,61,8,63),(9,15,11,13),(10,14,12,16),(17,23,19,21),(18,22,20,24),(25,31,27,29),(26,30,28,32),(33,37,35,39),(34,40,36,38),(45,51,47,49),(46,50,48,52),(53,59,55,57),(54,58,56,60)]])
 

32 conjugacy classes

class 1 2A···2G2H2I2J4A···4P4Q···4U
order12···22224···44···4
size11···14484···48···8

32 irreducible representations

dim111111111111112244
type++++++++++++++++-
imageC1C2C2C2C2C2C2C2C2C2C2C2C2C2D4C4○D42+ 1+42- 1+4
kernelC23.615C24C23.7Q8C23.8Q8C23.23D4C23.63C23C24.C22C23.65C23C23.67C23C23.10D4C23.78C23C23.Q8C23.11D4C2×C4.4D4C2×C42⋊2C2C22⋊C4C2×C4C22C22
# reps111111111122114822

Matrix representation of C23.615C24 ►in GL6(𝔽5)

120000
040000
004000
000100
000040
000004
,
400000
040000
001000
000100
000010
000001
,
100000
010000
004000
000400
000010
000001
,
100000
010000
001000
000100
000040
000004
,
120000
440000
004000
000400
000040
000001
,
300000
030000
000200
002000
000001
000010
,
300000
220000
000100
001000
000040
000004

G:=sub<GL(6,GF(5))| [1,0,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1,0,0,0,0,0,0,1],[1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,1,0,0,0,0,0,0,4,0,0,0,0,0,0,4],[1,4,0,0,0,0,2,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,4,0,0,0,0,0,0,1],[3,0,0,0,0,0,0,3,0,0,0,0,0,0,0,2,0,0,0,0,2,0,0,0,0,0,0,0,0,1,0,0,0,0,1,0],[3,2,0,0,0,0,0,2,0,0,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,0,0,4,0,0,0,0,0,0,4] >;
 

C23.615C24 in GAP, Magma, Sage, TeX 

C_2^3._{615}C_2^4
 
% in TeX
 
G:=Group("C2^3.615C2^4");
 
// GroupNames label
 
G:=SmallGroup(128,1447);
 
// by ID
 
G=gap.SmallGroup(128,1447);
 
# by ID
 
G:=PCGroup([7,-2,2,2,2,-2,2,2,253,120,758,723,268,1571,346,80]);
 
// Polycyclic
 
G:=Group<a,b,c,d,e,f,g|a^2=b^2=c^2=d^2=1,e^2=g^2=b,f^2=c*b=b*c,e*a*e^-1=a*b=b*a,f*a*f^-1=a*c=c*a,a*d=d*a,g*a*g^-1=a*b*c,b*d=d*b,g*e*g^-1=b*e=e*b,b*f=f*b,b*g=g*b,c*d=d*c,c*e=e*c,c*f=f*c,c*g=g*c,f*e*f^-1=d*e=e*d,d*f=f*d,d*g=g*d,f*g=g*f>;
 
// generators/relations
 

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